How do you tell the deputy prime minister that he's got his sums wrong and politely ask him to stop exaggerating about the local council?

That was the task assigned to Sheffield city council chief executive John Mothersole by Labour councillors, annoyed that Nick Clegg has been dissing the council over its spending on accommodation, consultants and its financial reserves.

In a letter to Clegg – which has conveniently found its way into the public domain – Mothersole said he has been "instructed to write" to Clegg in order to deal with "recent inaccuracies and misrepresentations with regard to various aspects of council expenditure".

Mothersole says there have been "many recent references made to spending of in the region of £2m on 'town hall meeting rooms'", which is "emphatically not the case".

While there is a £600,000 renovation of the town hall planned it is for general maintenance and to "boost income as it is used increasingly for weddings, civil ceremonies, registration of births etc" Mothersole says. "I am sure that all will support additional income being earned."

The letter also gently chides Clegg about his criticism of the use of consultants. "I note that a figure has recently been quoted as such expenditure being £4.7m," says Mothersole. "This is not the case. The actual spend approved on consultancy over the period referred to is £543,000."

In an open letter to Labour council leader Julie Dore last month, Clegg wrote: "The people of Sheffield deserve to know, as you know, some of the wasteful spending you have agreed to – such as paying dozens of employees to carry out trade union duties instead of doing their job and allocating more than £2.2m for town hall meeting room refurbishments.

He added: "They also deserve to know that in the last financial year the council showed a budget surplus of £6.2m and the latest statement of accounts shows £167m of 'useable reserves'. It is inexplicable that you have allowed this investment to sit in the council's coffers when it should be used to protect frontline services."

Mothersole's letter to Clegg says councillors are "concerned that the reserve figure quoted in various places is significantly higher than what in practice would be the truly useable cash reserve."

While the council's statement of accounts shows a figure of £167m, this is not the same as the "useable reserves", which are only £11m. "Again, the view of the council is that to quote a higher figure … would give a misleading impression that there is substantial cash available for expenditure without consequence. This is not the case."

A masterclass in gentle correction, Mothersole's letter concludes: "I hope this letter helps clarify these three key issues, which, as I have stated earlier, have been portrayed inaccurately on a number of occasions."

•Kevin Meagher is associate editor of Labour Uncut. He contributes to Left Foot Forward and is also a communications consultant. He tweets @KevinPMeagher